9 MEASURABLE FUNCTIONS 151 22. Let X be a locally compact space, Y a compact space, p a positive measure on X, and (fn)nzo a sequence of /x-measurable mappings from X to Y. Show that there exists a mapping G?, jc)i—>O-P(JC) of N x X into N with the following properties: (1) for each integer p ^ 0 and each integer n J> 0, the set ctpl(n) is /u-measurable; (2) for each x e X, the sequence (ftTp(x)(x))p^Q converges in Y. (One method is as follows. Taking a distance function on Y, define universally measurable subsets As of Y satisfying conditions (i) and (ii) of Section 4.2, Problem 3(a) and such that (in the same notation) AS'nAS" = 0. Then define subsets Bs of X inductively as follows: if Bs has been defined in such a way that, for each x e Bs, infinitely many terms of the sequence (/„(*)) belong to A,, we define Bs- as the set of all points x e Bs such that infinitely many terms of the sequence (fn(x)) belong to A,', and we define Bs« to be the com- plement of BS' in Bs. Now let x e X and let (ep)p^0 be the sequence of O's and 1's such that, if sp — (£t)o^t^P > then x e BSp for all p. Define crp(x) by induction as follows: a0(x) is the smallest integer n such that/n(x) e AS(), and ap(x) is the smallest integer n > op-i(x) such that fn(x) e A5p. Use Problem 21 to verify that this definition of ap(x) satisfies the conditions of the problem.) 23. Let ju, be a bounded positive measure on X and let/5:; 0 be a /x-integrable function. Show that there exists a lower semi continuous function g such that g^l/f (with the convention that 1/0= -foo) and such that gf is integrable (with the convention that the product of 0 and -f oo is 0). (Reduce to the case where/is bounded; consider the sets An of points x e X such that f(x) g l/«, and apply (13.7.9) to these sets to construct g.) 24. Let X, Y be locally compact spaces, p, a positive bounded measure on X, and TT a /x-measurable mapping from X to Y. For each/e JTR(Y), show that /° TT is jii-integ- rable and that the mapping /h-» (/o rr) dp, is a positive measure v on Y (called the image of JJL under TT, and denoted by ^(/i)). Generalize the results of Section 13.8, Problem 12, and Section 13.9, Problems 10 to 14. 25. Let X be a compact space and let (-07,,) be a sequence of finite partitions of X consisting of integral sets, such that wm is finer then wn whenever m > n (Problem 7). An ele- mentary martingale relative to the sequence (wn) is by definition a sequence (/„) of integrable functions ^ 0 such that: (i) /„ is constant on each set of the partition wn; (ii) If m > n, for each set A e wn we have (in other words, on each set A e w» the value of/„ is equal to the average of/m over A), (a) Let <a, b be real numbers such that 0 < a < b, and let Eflb be the integrable set of points x e X such that lim inf fn(x) < a< b < lim sup/n(jc). n-froo n-*oo Show that Eab is negligible. (Suppose if possible that p,(Eab) > 0. For each integer p > 0, let Fp be the union of the sets A e wn (n^p) such that A n Eab ^ 0; if F = P) Fp, then ^(F) ^ p*(Eat)) > 0. Show that for each integer/? and each A e wn (n *Zp) p such that A n Efli, 7^ 0, we have b * ft(A n F^) ^ a - p,(A) for all q J> w, by using the exists a sequence (tn) of finite real numbers such that £ |fB| = -|- oo